Auxiliary Gray-Scott Images and Movies

This is a collection of Gray-Scott images and movies that serve as a
supplement to the main exhibit. Some of these are
produced with parameters that are within the area covered by the main
exhibit, but are "in between" the standard grid points. Others are at
standard grid points but were produced with special starting patterns.

The parameters (F=0.098, k=0.056)

These parameters (and nearby values) support a very wide range of
behaviours and patterns. Although the final states are similar,
behaviour varies greatly depending on the type of initial pattern
(mainly red or mainly blue) and precise details of its configuration.

Blue spots grow into ovals, which readily elongate into
slowly-growing "wide worms" (upper-right) which maintain a fixed
spacing when traveling side by side (e.g. the pair in upper-left; and at 1:00
note the worm in the lower-left bends to the left to keep its distance from the
other two). When pressed from the side (center of screen at 1:28)
these wide worms narrow, and the narrowing travels quickly to both
ends, but the ends themselves remain wide. If a worm tip is
constrained it becomes narrow (watch center at 1:44) and the narrow
worm-tip grows more quickly. Ultimately the entire space is filled
with the narrow type of worm.

A large blue area will grow from the "corners", and the center also
steadily grows. The corners generally grow faster; the three seen here
are like the "wide worms" above. As above, they can bend (0:30 - 0:45)
but when poked from the side, transform into normal-width worms
(lower-left, 0:52). At 1:06, one of the tips narrows and then begins
lengthening much more quickly; this soon precipitates narrowing of the
other features and the large blue area shrinks to nearly nothing (1:25
- 1:50). A fourth wide-worm arises at 2:00, and the space eventually
becomes filled with parallel stripes. Worms can shorten and disappear,
as seen from 3:10-3:22. The drifting motion seen at the end of the
video continues indefinitely.

In a mostly blue background, most red spots become stable negatons.
However, any sufficiently convoluted red area, or an area containing a
blue "island", results in worm tips. These grow steadily, but twist
and turn in response to the negatons and other obstacles. The red
boundaries naintain a fairly constant width, and negatons expand a bit
when a red boundary touches them (right of center, 0:25-0:35).
Negatons will disappear if pressured (near center, 0:51).

(old video: P8pdhVBrsZQ was encoded poorly.)

Parameters near Uskate World

Here are some movies showing what happens at values of F and k close to
uskate world.

Accelerating time-lapse; speed doubles every 9 seconds. Space quickly
fills with concentric loops and negatons; the loops coalesce and
lengthen, becoming generally straighter and more parallel. Worm tips
grow until something makes them run into a wall, then another worm tip
grows in another direction (see 0:50 - 1:20). This process seems to
continue indefinitely: this time-lapse shows 2.44×109tu of
evolution.

(old video: XPndrF4-atQ was encoded poorly.)

Uskate World in One Dimension

Stable moving patterns also exist in one-dimensional Gray-Scott
systems, and at the same F and k parameter values as in the
2-D system.

This is a one-dimensional simulation; the colored bar shows the
pattern in the same colour-scheme as my 2-D movies; and the black and
white lines show the levels of u and v respectively. These F and
k parameter values support bound groups of negative solitons; this
particular pattern (called "1011") drifts to the right very slowly.

Accelerating time-lapse; speed doubles every 9.25 seconds. Solitons
and a few worms that grow to fill the space. The worms corral the
solitons but homotopy is preserved. The orientation of the worm-tips
drives the overall direction of movement of the pattern; when space is
full even the down-facing worm tips are forced to reverse direction.
(as seen at 1:30-1:40)

This behavior is typical of many parameters that support both worms
and solitons, and where the surface tension is positive.